In Sec. III, the cosmological perturbations of mLQC-I and mLQC-II are studied. We discuss initial
conditions and the subtle issue of the ambiguity in the choice of the variables \(\pi _a^{-2}\)  and \(\pi _a^{-1}\)  (present in the effective potential), which correspond to the quadratic and linear
inverse of the momentum conjugate to the
scale factor. In addition, to understand the issue of initial conditions
properly, we first introduce the comoving Hubble radius \(\lambda _H^2\)  and then state clearly how this is resolved in GR [cf. Fig. REF ],
and which are the relevant questions in mLQC-I [cf. Fig. REF ] and mLQC-II [cf. Fig. REF ]. From these figures it is clear that the BD vacuum cannot be
consistently imposed at the bounce It should be noted that anisotropies rise during
the contracting phase and generically dominate the earliest stages of the post-bounce of the homogeneous universe [1]}, [2]}, [3]}, [4]}. So, cautions must be taken,
when imposing initial conditions at the bounce., as now some modes are inside the (comoving) Hubble radius while others not. However, the fourth-order adiabatic vacuum may be imposed at this moment for both of these
two modified LQC models, as that adopted in LQC [5]}. In addition, in mLQC-I the de Sitter state given by Eq.(REF ) To be distinguished from the BD vacuum
described by Eq.(REF ) we refer to the state described by Eq.(REF ) as the de Sitter state. The difference between them is due to the term \(i/(k\eta )\) , which is not negligible in the deep contracting phase of the de Sitter background, as now \(|k\eta |\)  could be very small. For more details, see the discussions given in Sec. III.A, especially the paragraph after Eq.(REF ). can be imposed in the contracting phase as long as \(t_0\)  is sufficiently early, so the universe is
well inside the de Sitter phase. On the other hand, in mLQC-II and LQC, the BD vacuum can be imposed in the contracting phase as long as \(t_0\)  is sufficiently early, so
the universe becomes so large that the spacetime curvature is very small, and particle creation is negligible. With these in mind, the power spectra obtained in the three models, mLQC-I, mLQC-II and LQC, within the framework of the dressed metric approach were calculated and compared by imposing the initial conditions in the contracting phase. In particular, the spectra can be universally divided into three regimes, the infrared, intermediate and UV.
In the infrared and intermediate regimes, the relative differences in the amplitudes of the spectra can be as large as \(100\%\)  between mLQC-I and mLQC-II (the same is also true between mLQC-I and LQC), but
in the UV regime such differences get dramatically reduced, which is no larger than \(0.1\%\) . Since the modes in the UV regime are the relevant ones to the current observations and also their corresponding power spectra are scale-invariant, so these three models are all consistent with observations.
